×

Supercongruences for polynomial analogs of the Apéry numbers. (English) Zbl 1442.11039

The purpose of this article is to study generalizations of the \(q\)-Apery numbers defined by \[ A_ q (n) =\sum_{k=0}^nq^{(n-k)^2}\binom{n}{k}_q^2\binom{n+k}{k}_q^2. \] Let \(\Phi_m(q)\) denote the \(m\)th cyclotomic polynomials. The main theorem states a general congruence \((\text{mod}\Phi_m(q)^3)\), which implies a congruence for the \(q\)-Apery numbers as a special case.

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
05A30 \(q\)-calculus and related topics
11B83 Special sequences and polynomials
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Adamczewski, Boris; Bell, Jason Pierre; Delaygue, \'Eric; Jouhet, Fr\'ed\'eric, Congruences modulo cyclotomic polynomials and algebraic independence for \(q\)-series, S\'em. Lothar. Combin., 78B, Art. 54, 12 pp. (2017) · Zbl 1405.11019
[2] Ahlgren, Scott; Ekhad, Shalosh B.; Ono, Ken; Zeilberger, Doron, A binomial coefficient identity associated to a conjecture of Beukers, Electron. J. Combin., 5, Research Paper 10, 1 unnumbered page pp. (1998) · Zbl 0885.05017
[3] Ahlgren, Scott; Ono, Ken, A Gaussian hypergeometric series evaluation and Ap\'ery number congruences, J. Reine Angew. Math., 518, 187-212 (2000) · Zbl 0940.33002
[4] Almkvist, Gert; van Straten, Duco; Zudilin, Wadim, Generalizations of Clausen’s formula and algebraic transformations of Calabi-Yau differential equations, Proc. Edinb. Math. Soc. (2), 54, 2, 273-295 (2011) · Zbl 1223.33007
[5] Amdeberhan, Tewodros; Tauraso, Roberto, Supercongruences for the Almkvist-Zudilin numbers, Acta Arith., 173, 3, 255-268 (2016) · Zbl 1360.11004
[6] Andrews, George E., \(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenholme and Glaisher, Discrete Math., 204, 1-3, 15-25 (1999) · Zbl 0937.05014
[7] Ap\'ery, Roger, Irrationalit\'e de \(\zeta2\) et \(\zeta3\), Ast\'erisque, 61, 11-13 (1979) · Zbl 0401.10049
[8] Beukers, Frits, Some congruences for the Ap\'ery numbers, J. Number Theory, 21, 2, 141-155 (1985) · Zbl 0571.10008
[9] Beukers, Frits, Another congruence for the Ap\'ery numbers, J. Number Theory, 25, 2, 201-210 (1987) · Zbl 0614.10011
[10] Chan, Heng Huat; Cooper, Shaun; Sica, Francesco, Congruences satisfied by Ap\'ery-like numbers, Int. J. Number Theory, 6, 1, 89-97 (2010) · Zbl 1303.11009
[11] Chan, Heng Huat; Zudilin, Wadim, New representations for Ap\'ery-like sequences, Mathematika, 56, 1, 107-117 (2010) · Zbl 1275.11035
[12] Chowla, Sarvadaman; Cowles, John; Cowles, Mary, Congruence properties of Ap\'ery numbers, J. Number Theory, 12, 2, 188-190 (1980) · Zbl 0428.10008
[13] Chu, Wenchang, A binomial coefficient identity associated with Beukers’ conjecture on Ap\'ery numbers, Electron. J. Combin., 11, 1, Note 15, 3 pp. (2004) · Zbl 1062.05016
[14] coster-sc Matthijs J. Coster, Supercongruences, Ph.D. thesis, Universiteit Leiden, 1988.
[15] D\'esarm\'enien, Jacques, Un analogue des congruences de Kummer pour les \(q\)-nombres d’Euler, European J. Combin., 3, 1, 19-28 (1982) · Zbl 0485.05006
[16] Deutsch, Emeric; Sagan, Bruce E., Congruences for Catalan and Motzkin numbers and related sequences, J. Number Theory, 117, 1, 191-215 (2006) · Zbl 1163.11310
[17] fs-qbinomial Sam Formichella and Armin Straub, Gaussian binomial coefficients with negative arguments, Preprint (2018), http://arxiv.org/abs/1802.02684arXiv:1802.02684. · Zbl 1433.05016
[18] Gessel, Ira, Some congruences for Ap\'ery numbers, J. Number Theory, 14, 3, 362-368 (1982) · Zbl 0482.10003
[19] gorodetsky-cong-q Ofir Gorodetsky, \(q\)-congruences, with applications to supercongruences and the cyclic sieving phenomenon, Preprint (2018), http://arxiv.org/abs/1805.01254arXiv:1805.01254. · Zbl 1423.11043
[20] Granville, Andrew, Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers. Organic mathematics, Burnaby, BC, 1995, CMS Conf. Proc. 20, 253-276 (1997), Amer. Math. Soc., Providence, RI · Zbl 0903.11005
[21] gz-rama-q Victor J. W. Guo and Wadim Zudilin, A \(q\)-microscope for supercongruences, Preprint (2018), http://arxiv.org/abs/1803.01830arXiv:1803.01830. · Zbl 1464.11028
[22] Jouhet, Fr\'ed\'eric; Mosaki, Elie, Irrationalit\'e aux entiers impairs positifs d’un \(q\)-analogue de la fonction z\^eta de Riemann, Int. J. Number Theory, 6, 5, 959-988 (2010) · Zbl 1204.11110
[23] Kac, Victor; Cheung, Pokman, Quantum calculus, Universitext, x+112 pp. (2002), Springer-Verlag, New York · Zbl 0986.05001
[24] Krattenthaler, Christian; Rivoal, Tanguy; Zudilin, Wadim, S\'eries hyperg\'eom\'etriques basiques, \(q\)-analogues des valeurs de la fonction z\^eta et s\'eries d’Eisenstein, J. Inst. Math. Jussieu, 5, 1, 53-79 (2006) · Zbl 1089.11038
[25] MacMahon, Percy A., Combinatory analysis, Two volumes (bound as one), xix+302+xix+340 pp. (1960), Chelsea Publishing Co., New York · Zbl 0101.25102
[26] Mimura, Yoshio, Congruence properties of Ap\'ery numbers, J. Number Theory, 16, 1, 138-146 (1983) · Zbl 0504.10007
[27] Olive, Gloria, Generalized powers, Amer. Math. Monthly, 72, 619-627 (1965) · Zbl 0215.07003
[28] Osburn, Robert; Sahu, Brundaban, Congruences via modular forms, Proc. Amer. Math. Soc., 139, 7, 2375-2381 (2011) · Zbl 1236.11046
[29] Osburn, Robert; Sahu, Brundaban; Straub, Armin, Supercongruences for sporadic sequences, Proc. Edinb. Math. Soc. (2), 59, 2, 503-518 (2016) · Zbl 1411.11005
[30] Pan, Hao, A generalization of Wolstenholme’s harmonic series congruence, Rocky Mountain J. Math., 38, 4, 1263-1269 (2008) · Zbl 1236.11019
[31] van der Poorten, Alfred, A proof that Euler missed \(\ldots\) Ap\'ery’s proof of the irrationality of \(\zeta(3)\), Math. Intelligencer, 1, 4, 195-203 (1978/79) · Zbl 0409.10028
[32] Shi, Ling-Ling; Pan, Hao, A \(q\)-analogue of Wolstenholme’s harmonic series congruence, Amer. Math. Monthly, 114, 6, 529-531 (2007) · Zbl 1193.11018
[33] Straub, Armin, A \(q\)-analog of Ljunggren’s binomial congruence. 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), Discrete Math. Theor. Comput. Sci. Proc., AO, 897-902 (2011), Assoc. Discrete Math. Theor. Comput. Sci., Nancy · Zbl 1355.05053
[34] s-apery Armin Straub, Multivariate Ap\'ery numbers and supercongruences of rational functions, Algebra & Number Theory 8 (2014), no. 8, 1985-2008. · Zbl 1306.11005
[35] Zheng, De-Yin, An algebraic identity on \(q\)-Ap\'ery numbers, Discrete Math., 311, 23-24, 2708-2710 (2011) · Zbl 1242.05007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.